Topographical Tools for Filtering and Segmentation 1

Watersheds on Node- Or Edge-Weighted Graphs
1. Auflage 2019
ISBN: 978-1-78630-157-4
Verlag: Wiley

Buch, Englisch, 320 Seiten, Format (B × H): 163 mm x 239 mm, Gewicht: 612 g

Mathematical morphology has developed a powerful methodology for segmenting images, based on connected filters and watersheds. We have chosen the abstract framework of node- or edge-weighted graphs for an extensive mathematical and algorithmic description of these tools.

Volume 1 is devoted to watersheds. The topography of a graph appears by observing the evolution of a drop of water moving from node to node on a weighted graph, along flowing paths, until it reaches regional minima. The upstream nodes of a regional minimum constitute its catchment zone.

The catchment zones may be constructed independently of each other and locally, in contrast with the traditional approach where the catchment basins have to be constructed all at the same time. Catchment zones may overlap, and thus, a new segmentation paradigm is proposed in which catchment zones cover each other according to a priority order. The resulting partition may then be corrected, by local and parallel treatments, in order to achieve the desired precision.

Autoren/Hrsg.

Meyer, Fernand

Inhaltsverzeichnis

Notations xiii

Introduction xxvii

Part 1. Getting Started 1

Chapter 1. A Primer to Flooding, Razing and Watersheds 3

1.1. Topographic reliefs and topographic features 3

1.1.1. Images seen as topographic reliefs and inversely 3

1.1.2. Topographic features 5

1.1.3. Modeling a topographic relief as a weighted graph 8

1.2. Flooding, razing and morphological filters 10

1.2.1. The principle of duality 10

1.2.2. Dominated flooding and razing 10

1.2.3. Flooding, razing and catchment zones of a topographic relief 16

1.3. Catchment zones of flooded surfaces 18

1.3.1. Filtering and segmenting 18

1.3.2. Reducing the oversegmentation with markers 19

1.4. The waterfall hierarchy 26

1.4.1. Overflows between catchment basins 26

1.5. Size-driven hierarchies 28

1.6. Separating overlapping particles in n dimensions 31

1.7. Catchment zones and lakes of region neighborhood graphs 33

1.8. Conclusion 37

Chapter 2. Watersheds and Flooding: a Segmentation Golden Braid 39

2.1. Watersheds, offsprings and parallel branches 40

2.2. Flooding and connected operators 43

2.3. Connected operators and hierarchies 45

2.4. Hierarchical segmentation: extinction values 47

Chapter 3. Mathematical Notions 49

3.1. Summary of the chapter 49

3.2. Complete lattices 49

3.2.1. Partial order and partially ordered sets 49

3.2.2. Upper and lower bounds 50

3.2.3. Complete lattices 50

3.2.4. Dyadic relations on a complete lattice 51

3.3. Operators between complete lattices 51

3.3.1. Definition of an operator 51

3.3.2. Properties of the operators 52

3.3.3. Erosion and dilation 52

3.3.4. Opening and closing 53

3.4. The adjunction: a cornerstone of mathematical morphology 53

3.4.1. Adjoint erosions and dilations 53

3.4.2. Increasingn

Über Autor(innen)

Fernand Meyer has been working at the Center for Mathematical Morphology of MINES ParisTech since 1975. He participated actively in the development of mathematical morphology, particularly in the field of segmentation and filtering.